CQI Academy

Section I · 19 questions · Apply

Technical mathematics

The exam expects shop math you can do with a pencil and a TI-30 (a common non-programmable scientific calculator). Fractions, algebra, geometry, right-triangle trig, unit conversions, scientific notation, and basic descriptive statistics. Show units. Box the number before you look at the choices.

I.A Basic shop math Apply

You add, subtract, multiply, and divide fractions and decimals. You square numbers and take square roots. You round and truncate. You compute percent change.

PERCENT CHANGE percent change = (new − original) / original × 100 A drop from 50 to 40 is (40−50)/50 × 100 = −20%

Fractions

Find a common denominator to add or subtract. Invert and multiply to divide. Reduce the answer. Mixed numbers: 3 1/4 = 13/4. Keep the original units.

Example

A stock allowance is 3/8" and you already removed 5/32". Remaining = 3/8 − 5/32 = 12/32 − 5/32 = 7/32".

Rounding versus truncating

Rounding looks at the next digit: 5 or more goes up. Truncating chops extra digits with no rounding. Significant digits are the digits that carry meaning given the instrument. A 0–1" micrometer reading to 0.0001" should not be written as 0.43750000 if the instrument cannot support it.

Negative numbers follow the same rounding rules on the absolute value, then the sign returns. −1.26 rounded to one decimal is −1.3.

I.B Basic algebra Apply

First-degree, one unknown. Isolate the variable by doing the same operation to both sides.

SOLVE 3x + 7 = 22 3x = 15 x = 5 Tolerance buildup (simple): total = A + B + C If two lengths are known, the third is total minus the knowns.

I.C Basic geometry Apply

Shape	Need-to-know
Rectangle	Area = L × W · Perimeter = 2(L+W)
Triangle	Area = ½bh · Angles sum to 180°
Circle	C = πd = 2πr · A = πr² · r = d/2
Right circular cylinder	Volume = πr²h · Lateral area = 2πrh
Rectangular solid	Volume = LWH · Surface = 2(LW+LH+WH)
Sphere	Volume = 4/3 πr³ · Surface = 4πr²

Complementary angles add to 90°. Supplementary angles add to 180°. A line is fixed by 2 points. A circle is fixed by 3 non-collinear points (or center + radius). The exam may ask the minimum number of coordinate points to define a shape.

USE π = 3.1416 UNLESS TOLD OTHERWISE A 2.000" diameter bar, cross-section area = π(1.000)² = 3.1416 in²

I.D Basic trigonometry Apply

Right triangles only, for this exam. SOH-CAH-TOA and Pythagoras.

RIGHT TRIANGLE sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent a² + b² = c² (c is the hypotenuse) Sine bar rise = sine(angle) × center distance A 5.000" sine bar at 30° needs a stack of 5.000 × 0.5000 = 2.500"

Inverse functions (sin⁻¹) give the angle when you know two sides. Set the calculator to degrees, not radians, unless the problem says otherwise.

I.E Measurement systems Apply

Convert inside English, inside SI (the international metric system), and across the two. Write the factor so units cancel.

From	To	Multiply by
1 inch	mm	25.4 exactly
1 inch	micron (µm)	25 400
1 microinch	µin	0.0254 µm
1 foot	inch	12
1 meter	mm	1 000
1 mm	µm	1 000
1 kg	lb (mass, approx.)	2.2046
°C to °F		°F = (9/5)°C + 32
°F to °C		°C = (°F − 32) × 5/9

Example

0.0004 inch in microns: 0.0004 × 25.4 × 1000 = 10.16 µm.

I.F Numeric conversions Apply

Scientific notation: 0.00025 = 2.5 × 10⁻⁴. 12 400 = 1.24 × 10⁴. Fractions to decimals: divide. Decimals to fractions: put over a power of ten and reduce. 0.125 = 125/1000 = 1/8.

I.G Basic statistics Apply

Moved into Section I in the 2025 BoK (Body of Knowledge). You must calculate, not just define.

DESCRIPTIVE STATS mean (x) = sum of values / n median = middle value of the ordered list (average the two middles if n is even) mode = most frequent value (a set can have none, one, or many) range (R) = max − min variance (sample) s² = Σ(x − x)² / (n − 1) std. dev. s = √variance proportion = count in category / n percentage = proportion × 100

Use n − 1 in the sample variance denominator unless the problem clearly gives a complete population (then divide by N). Exam items are almost always samples.

Normal distribution: symmetric bell, mean = median = mode, about 68% of values within ±1s, 95% within ±2s, 99.7% within ±3s (empirical rule). The exam asks you to recognize those traits and to compute a simple z if they hand you the pieces: z = (x − μ) / σ.

Graphs they name: scatter diagram (relationship between two variables), tally / check sheet (counts as you inspect), bar chart (compare categories). A histogram is in Section IV tools but you should still know it is the distribution picture of one variable.